1
vote
2answers
50 views

Question about a notation. Norm of the derivative of a function at a point

Given is an analytic function from $M$ to $N$, both equipped with conformal Riemannian metric, say $g$ and $h$ resp. What might the $h$ norm of the derivative of the function at a point mean? ...
4
votes
0answers
73 views

universal covering of punctured plane and Poincaré metric

I want to prove the following result: Let $\Omega$ be the domain $\mathbb{C}\backslash \{a_1,a_2,a_3,a_4\}$. Its universal covering is the unit disk, and the standard Poincaré metric is pulled back to ...
3
votes
1answer
79 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
4
votes
2answers
60 views

Orientation on Riemann surfaces

$\mathcal{X}$ is a Riemann surface and $\mathcal{E}^{(2)}(\mathcal{X})$ is the $\mathbb{C}$-Vector space of all differentiable $2$-forms on $\mathcal{X}$. I want to define the orientation of ...
0
votes
2answers
75 views

Homotopy invariance of line integral on manifolds

Consider a 1-form: $\omega\in\Gamma(\mathrm{T}^*M)$ and two differentiable curves: $\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$ together with a ...
5
votes
2answers
112 views

How many differential forms on the complex plane?

I am puzzled by the fact that the two differential forms $$\begin{array}{cc} dz=dx+i dy , & d\overline{z}=dx-i dy \end{array} $$ are $\mathbb{C}$-linearly independent, even if the underlying ...
3
votes
0answers
26 views

Can anyone prove this identity without passing through the complexified tangent space?

Let $\rho: \mathbb{C} \to \mathbb{R}$ be a smooth function, $\Omega = \{ z : \rho(z) <0 \}$, and suppose $|\nabla \rho| = 1$ on $b\Omega$. It is true that $$\int_{b\Omega} f(z) d\bar{z} = -2i ...
1
vote
2answers
102 views

What does $d\bar z$ mean?

What does $d\bar z$ mean? For a manifold, given a local coordinate, $dx$ acts on tangent vectors and gives its corresponding components. What does $d\bar z$ do? The complex field is a one ...
2
votes
1answer
70 views

Differential form on complex torus

Suppose $T= \mathbb{C}^n/\Gamma$ is a $n$-dim complex torus. How to prove that every exact $2$-form which has no $(0,2)$ component must be the image of a $(1,0)$ form? Is every torus Kahler? If the ...
6
votes
1answer
128 views

Rank of a jet bundle of a vector bundle.

I am trying to understand the jet bundles but currently I am stuck on the following questions: Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...
0
votes
1answer
22 views

Rewriting line integral for complex-valued function

Context: Suppose $f = \phi + i\psi$ is continuous and $\gamma(t):[a, b] \to \mathbb{C}$ is a curve. Then we define the integral of $f$ along $\gamma$ to be $$ \int_\gamma\!f = \int_a^b ...
0
votes
1answer
53 views

Geometric interpretation of $\partial/\partial z$

My understanding is that analytic derivative ,$\partial\phi/\partial z$, and anti-analytic derivative ,$\partial\phi/\partial\bar{z}$, are resp. tangent and normal to the curve $\phi$. Am i right?can ...
2
votes
0answers
61 views

Interpretation of a line integral in complex analysis

$\newcommand{\C}{\mathbb{C}}$ Suppose $f\colon \Omega\subset \C\to\C$ is a holomorphic function and $\gamma:[0,1]\to\Omega$ is a continuous path. If $\Omega=\C\setminus\{0\}$, $\gamma(t):= e^{2\pi i ...
0
votes
0answers
31 views

Reality condition on a metric

I'm studying a coördinate-transformation on a 2n-dimensional real manifold, where I can locally define the coördinates as $(x^1,...,x^{2n})\in\mathbb{R}^{2n}$, and transform them to: ...
0
votes
1answer
58 views

definition of a map from $CP^1$

I think this is a very easy question, but I've got problems understanding how the function in the second exercise of this pdf (that I found online on google and I wanted to try in order to improve my ...
2
votes
0answers
60 views

Inverse Function Theorem (results using it)

Hi i'm thinking in some ideas for my bachelor thesis. I'm working in a more "general" framework than manifolds, and i found that the Inverse Function Theorem is valid in such structures. So i was ...
1
vote
1answer
56 views

holomorhicity implies harmonic function in several variables

I had read somewhere that it follows by cauchy riemann equations that any holomorphic or anti-holomorphic function $f$ from an open subset of $C^n$ to $C$ is harmonic i.e $\sum_{i=1}^n ...
2
votes
1answer
285 views

Closed but not exact one-form on $S^2$

I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3. ...
7
votes
1answer
230 views

Möbius transformation in the complex plane.

Assume that $U$ be a line in the complex plane. And assume a Möbius transformation $\phi $ sends $ U $ again to a line. How can I classify all such $\phi$? I want to write my ideas. But, I ...
0
votes
3answers
138 views

top journals in analysis

as an undergraduate I find analysis as my favorite.I want to read journals regarding that. give me top 5 journals in analysis(real,complex)? top 5 journals in differential geometry? and generally some ...
5
votes
0answers
166 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
1
vote
1answer
60 views

Computing some differential forms using complex coordinates

I was computing some things in the Poincaré disk $\mathbb{H}^2$ in complex coordinates and then I tried to show that $\sigma_r(z) = \frac{r^2}{z}$ is an isometry. However $d\sigma_r = ...
3
votes
0answers
78 views

Explicitly realizing Riemann surfaces as a quotient of the upper-half plane

Let $\Sigma_g$ be a Riemann surface of genus $g \ge 2$. Then it is known that $\Sigma_g$ is (holomorphically) a quotient of the upper-half-plane (or unit disk) by a group $\Gamma$ of hyperbolic ...
2
votes
1answer
121 views

Construct a map from unit disk to upper half-plane

I want to construct this map in high-dimensional case. Let $D=\{x \in \mathbb{R}^n:|x|^2<1\}$,and $H=\{u\in\mathbb{R}^n:u^n>0\}$. Well, it is quite clear when $n=2$, but I find it is hard for me ...
4
votes
1answer
105 views

A complex structure on the tangent space

I am reading the book Riemann surface by Donaldson. I want to understand the following Lemma (p.74). Lemma. Let $X$ be a Riemann surface. There is a unique way to define a complex structure on ...
4
votes
1answer
95 views

Diffeomorphisms preserving harmonic functions

I'm looking for smooth maps $ \Bbb R^n \to \Bbb R^n $ with the property that, whenever $ h $ is a harmonic function ($\Delta h=0$), $ h\circ f $ is also harmonic. Is there a nice characterization of ...
1
vote
0answers
100 views

Use Möbius Transformation Normal Form to prove Lambda

I'm just completely lost on how to answer this question: Let $$\frac{Tz-p}{Tz-q}=\lambda \frac{z-p}{z-q}$$ be the normal form of a Möbius transformation with two fixed points. Prove that $\lambda$ = ...
0
votes
1answer
73 views

What is complex conjugation in Hyperbolic Geometry

Suppose we are working in the hyperbolic plane, that is the set $$ \mathbb{H}^2 = \{ u + iv \colon v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$ Now, formally, I can rewrite ...
4
votes
1answer
155 views

A step in the proof of the Riemann Mapping Theorem

Let $\Omega \subsetneq \Bbb C$ be open and simply connected. Let $\overline{\Bbb C}$ denote the Riemann sphere and assume without loss of generality that $0 \in \overline{\Bbb C} \backslash \Omega$. ...
4
votes
1answer
109 views

Prove that the circle $S^1$ is not the boundary of any compact manifold with boundary in $\mathbb R^2-{(0,0)}$

Suppose it were, then define a 1-form $w:=\frac{1}{x^2+y^2}(-y\,\mathrm dx+x\,\mathrm dy)$. Firstly , I try to evaluate $\int_{S^1}w$ by two ways . Firstly, let $F\colon[0,2 \pi]\to S^1$ defined by ...
1
vote
0answers
85 views

What happens to small squares in Riemann mapping?

I have a square $S$, and I want to convert it to the unit disc $D$. The Riemann mapping theorem says that I can to it with a conformal bijective map. But, any such mapping will cause some distortion. ...
1
vote
1answer
140 views

uniformization theorem - squares and circles

I am trying to understand the uniformization theorem and get some intuition about it. The uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of ...
4
votes
1answer
88 views

Geodesics of conformal metrics in complex domains

Let $U$ be a non-empty domain in the complex plane $\mathbb C$. Question: what is the differential equation of the geodesics of the metric $$m=\varphi(x,y) (dx^2+dy^2)$$ where $\varphi$ is a ...
7
votes
1answer
183 views

Is manifold mapping degree equal to algebraic degree for polynomials?

If $M$ and $N$ are oriented $n$-manifolds and $f: M \to N$ then the degree of $f$ is given by $$ \deg f = \sum_{p \in f^{-1}(q)} sign_p f $$ where $q$ is a regular value and the sign is $+1$ if $f$ is ...
2
votes
0answers
94 views

Schwarz Lemma in Differential Form

Suppose $w=f(z)$ is a conformal self map of $\mathbb{D}$. From Schwarz Pick Lemma we have $|\frac{dw}{dz}|=\frac{1-|w|^2}{1-|z|^2}$. Could any one explain me In differential form how this becomes ...
3
votes
1answer
131 views

smoothness of hopf fibration projection with respect to standard differentiable structure on unit sphere

We know that steographic projection defines a differentiable structure on $S^n$ by sending points on $S^n$ to hyperplane $\{x^{n+1}\}=0$. In fact, stereographic projection $\sigma_P: S^n- P\to R^n $ ...
2
votes
1answer
120 views

Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
9
votes
3answers
424 views

Where can I learn about complex differential forms?

So I'm a 3rd year grad student in number theory/modular forms/algebraic geometry, and I've worked with differential forms from an algebraic point of view without ever knowing what they really are. I'd ...
1
vote
1answer
61 views

Nadirashvili surface (part 3)

The article that I'm considering is 'Notes sur la démonstration de N. Nadirashvili des conjectures de Hadamard et Calabi-Yau' by Pascal Collin and Harold Rosenberg. In the proof of the appendix (of ...
2
votes
1answer
87 views

Independence of coordinate charts in the definition of the order of a pole of a meromorphic 1-form on a Riemann surface

I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One ...
1
vote
1answer
247 views

any two simply connected open set in the plane R^2 are diffeomorphic

Prove that any two simply connected open set in the plane R^2 are diffeomorphic. I know that in the complex plane any simply connected open set is diffeomorphic to either complex plane or open unit ...
0
votes
1answer
84 views

How to compute geodesics in upper half-plane?

Given two points in the upper half-plane with the usual hyperbolic metric, the geodesic between them is found by drawing a circle through them that crosses the real axis at right angles. However, if I ...
1
vote
1answer
95 views

Kaehler-Einstein metric on Calabi-Yau manifold

I am reading "Complex geometry" by D. Huybrecht. On p.223 the books says that "If $c_{1}(X)=0$, e.g. if the canonical bundle $K_{X}$ is trivial, and $g$ is Kaehler-Einstein metric, then Ric$(X,g)=0$. ...
3
votes
1answer
132 views

real part of a holomorphic function from a PDE

I have some problem that I can't figure out myself. Hope that someone can help me out. The problem is: Let $f : U \to \mathbb{R}$ be some real function on a simply-connected domain $U \subset ...
1
vote
1answer
129 views

Showing holomorphic functions are preserved under pullback by a holomorphic map

Let $f: X\rightarrow Y$ be a holomorphic mapping of complex manifolds and assume for simplicity that $dim(X)=dim(Y)=1$. I want to show that it preserves holomorphic functions under pullback. We define ...
1
vote
1answer
100 views

Definition of Holomorphic map of complex manifolds

Let $X,Y$ be complex manifolds and let $f: X \rightarrow Y$ be a continuous map. When exactly do we say that $f$ is "holomorphic"? I am interested in the basic definition (possibly using charts), not ...
2
votes
0answers
123 views

Conjugate function reverse angle in a neighborhood

Can you please help me with this question. Prove that if conjugate of $F$ is analytic in a neighborhood of $z_0$ in $C$ then $F$ reverses angles at $z_0$. Can anyone please explain that for me? what ...
4
votes
1answer
105 views

Specific homotopy between complex conjugation and the identity.

Consider the set $\mathcal{C} = C^{\infty}(\mathbb{C}^*, \mathbb{C}^*)$, where $\mathbb{C}^* = \mathbb{C}\backslash\{0\}$. Both $f(z) = z$ and $g(z) = \bar{z}$ can be seen as elements in ...
1
vote
1answer
71 views

a special extension of a two variable function

We consider the function $f(x,y)=x^2+y^2$ in $\omega = (0,1)^2.$ I am wondering about the existence of a $C^2-$extension $F$ of $f$ in $\Omega = (0,2)^2$ such that $F$ is harmonic in ...
6
votes
1answer
419 views

how to understand the tensor product canonical line bundle $\otimes$ dual bundle

Suppose we have a Riemann surface $M$ together with a holomorphic vector bundle $E \to M$ of rank n. let $K$ denote the canonical line bundle and let $E^*$ denote the dual bundle I am trying to ...